Legendre–Clebsch condition explained

In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum.

For the problem of minimizing

\int ^{b} _{a} L(t,x,x') dt .

the condition is

L _{x'} (t,x(t),x'(t)) \ge 0 , \forall t \in[a,b]

Generalized Legendre–Clebsch

In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition,^[1] also known as convexity,^[2] is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e.,

(\partial H)/(\partial u) = 0

The Hessian of the Hamiltonian is positive definite along the trajectory of the solution:

(\partial ^{2} H)/(\partial u ^{2}) > 0

In words, the generalized LC condition guarantees that over a singular arc, the Hamiltonian is minimized.

See also

• Bang–bang control

Further reading

• Book: Hestenes, Magnus R. . Magnus Hestenes

. Magnus Hestenes . Calculus of Variations and Optimal Control Theory . New York . John Wiley & Sons . 1966 . A General Fixed Endpoint Problem . 250–295 .

• Web site: Legendre condition. Encyclopedia of Mathematics. Springer.

Notes and References

1. H. M. . Robbins . A Generalized Legendre–Clebsch Condition for the Singular Cases of Optimal Control . IBM Journal of Research and Development . 11 . 4 . 361–372 . 1967 . 10.1147/rd.114.0361 .
2. Book: Choset, H.M. . 2005 . Principles of Robot Motion: Theory, Algorithms, and Implementation . The MIT Press . 0-262-03327-5.